I wouldn't worry about the tests, all they are trying to establish are your attitude, aptitude and basic intelligence which are all important, you say the physical part doesn't scare you, well it should because no matter how fit you are now, you will get fitter and constantly challenged and helped to improve your fitness. I know a lot of guys that have on paper and appearance looked superbly fit, they could lift heavy weights and run five miles every day etc etc.
However it's stamina and will power that will keep.you going. It's all very well doing a five mile run, but doing the same with webbing, bergan, rifle, ammo, boots and( if your instructor is cruel) helmet.
I would definitely be concerned about your existing back injury because you will be putting a lot of strain on it, not just carrying loads but simple things like sit ups. You can tell them about your injury and hope they let you join or hide it and hope for the best, however if they X-Ray it and discover a previous undeclared injury then they will.probably throw you out onto the street.
I wish you the best of luck, there are many training books out there about how to get fit for service, I recall one that I used and its about royal marine fitness training week by week. If you can stick by that regime things might be easier.
Once again good luck